The Boolean expression for the output \(f\) of the multiplexer shown below is

Key idea: the two select lines P and Q choose either R or its complement, so the output is an XOR of the three signals.

• Map select values to inputs: with select bits (P,Q) = 00 the top input R is chosen; 01 chooses R̅; 10 chooses R̅; 11 chooses R.

• Write the sum-of-products from the MUX selection:

  f = P'Q'R + P'QR' + PQ'R' + PQR

• Factor by R and R':

  f = R(P'Q' + PQ) + R'(P'Q + PQ')

• Recognize the grouped terms: P'Q' + PQ = (P ⊕ Q)' (XNOR) and P'Q + PQ' = P ⊕ Q. Thus f = R·(P ⊕ Q)' + R'·(P ⊕ Q) = R ⊕ (P ⊕ Q).

• Using associativity of XOR, the final simplified expression is:

  f = P ⊕ Q ⊕ R